Т.А. Леонтьева, В.С. Панферов, В.С. Серов - Задачи по теории функций комплексного переменного с решениями (1118152), страница 54
Текст из файла (страница 54)
, N,Nk> rk)}, kv,, = eia.+iNr2 . ln(z -ak) .mHHa 6eCKOHe'-!HOCTl1 -B11xpe11CTO'iHl1Kr _ = I := 1 rkHMe1Dmero Hac0611JibHOCTbl{).Haih11 3aKOH H3MeHeHHH s11xpeHCTO'-IH11Ka .a;11nonH, Haxo.a;muerocH Ba* *HJIH Ha 6eCKOHe<IHOCTl1 np11 cneny!Oll{HX 0.[(HOJIHCTHb!X KOHcpopM-HblX OT06pa)l(eHlUIX OKpeCTHOCTH 3THX TO'-!eKc, 0,0:c_,a) ~(z) =a+ C 1(z - a)+ ... ; 6) ~(z) =a+ Crfz + ... ;B) ~(z)= (C - l)/(z - a)+ C0 + ... ; r)Omeembl.~(z)= C 1z + C0 + ....IIp11 onHon11crnoM KOHcpopMHOM orn6pa)l(eHHH BHxpeHCT0'-1-HHK rrepexonHT B BHXpeHCTO'-IHHK TOH )Ke o6HJibHOCTH 11 f!HTeHCHBHOCTH.a) n11nOJib(a, p)->(a,B) n11nOJib(a, p)->(oo,pC,);·6) n11nonb (oo,p)->(a,pC,);r) n11nonh (oo,p)->(oo,p/C1).p/C1);KOMOJICKCHbIH OOTeHuuaJIB 3JICKTpOCTaTHKe H TennonpOBOLlHOCTHIInocKoe 3JieKrpocrnTwiecKoe none, T. e.
none seKTopa ttanpH/KeHHOCTHE = { Ex(x, y), Ey(x, y)} i!BJIHCTCH rroneM rpa.D,Hettrn E = -grad v(x, y),v(x, y) - noTeHu11an :rneKTpOCTaTH'iCCKOro nonil, T. e. none E noTCHl.(11anbHO. Bo BCCX TO'IKax, B KOTOpblX HCT 3apii,D;OB, 3nCKTpOCTaTH'-ICCKOC noneCOJICHOHJJ,aJlbHO . .lJ:nii ero 113Y'-ICHl1il uenecoo6pa3HO BOCnOJib30BaTbCil noHi!THCM KOMnneKcHoro n0Tettu11ana. KoAt11!leKCHbl.M nomeH£fUG!lOA1 nollflEH33b!B3IOT <i>YHKUHIOf(z)= u(x, y) + iv(x, y),rp;edv<l>yHKUHil=-[E,dx + E,dy], du= -E,dx + E,dy.u(x, y) -c11nosail <l>YHKU11il,v(x, y) -noTeHUHaJl 3JieKrpOCTa-TH'-!eCKOro noJlil,IEIJlHHHH ypoBHilnHHHH ypOBHil U(X,= lf'(z)I,Arg E= -[n/2 + Argf'(z)].v(x, y) Ha3b!B3IOTCil 3KGU/10111eHL/UG!lbHblA1Uy) - CUJ/OGblJ\IU Jll1HHi!Ml1, f C paBHa pa60TCnepeMemett1111 en11ttwrnoro 3apHna no 3aMKHYTOMY KOHTypy,JIHHl1i!M11,nOJiil npHNe(noToKBCKTOpa Harrpi!)KCHHOCTH) paBeH anre6paH'-!CCKOH cyMMe pacrronO)KCHHb!XBHYTPH KoHTypaC 3apii.D,OB,YMHO)KCHHOH Ha2rt.351I1Pl1JIO)l{EHVU! K MEXAHI1KE H <l>l13HKE1.8bIHCHl1Tb xapaKTep :rneKTpOCTaTl14eCKOro llOJrn, COOTBeTCTByIOmeroKOMnneKCHOMY n0Tem.111any f(z) = i In z.= -arg z, v = In lzl.Pewem1e.
utOTCH 0Kpy)f(ttOCT11 lzl =3Ks11noTem.111anbHbIM11 n11H11HMl1 HBAA-C 1, C11J10BbIM11 n11tt11HMl1HBAAIOTCH11y411 arg z = C2 ;IEI = lzl-1> Arg E =-(n/2 + Arg (i/2)) = Arg z -n.BeKTOp HanpH)f(eHHOCTl1 HanpasneH B}lOJJb Cl1JJOBOH JJl1Hl111 K Ha4any KOOpL\11HaT. Oco6oi1 T04Koi1 HBJJHeTCH T04Ka z = 0, B Hei1 pacnono1KeH 3apHJl.11Bb16epeM B Ka4ecrne KOHrypa C OKPYIKHOCTh lzl = r. Tor}la £ 11= -lzl- = -r- ,Ne= paBHafc -dsr= -2n. Be1111411Ha 3ap»na, pacnoJJO)f(eHHoro B T04Ke z = 0,-1.
0TCIO}la CJJe}lyeT, 4TO KOMnneKCHhlH llOTeHu11an, COOTBeTCT+ 1 B T04Ke z = 0, 11MeeT BHJlf{z) = -i In z.sy1om11if 3ap5IJlY2. BhrnCHHTb xapaKTep :rneKTpocrnT114eCKOro non», coorneTcrnyIOmeroI z.KOMnneKCHOMY noTeHu11any f(z) = iyxPeiuem1e. u(x, y) = - ,- - , , v(x, y) = - ,- , . 3KB11noTeHu11anhHhIM11x- + yx- + fOx, Kacat0m11ecH oc11Oy, c1111osh1M11 n11HIDIM11 - 0Kpy1KHocn1 c ueHTpOM Ha oc11 Oy, Kaca1om11ec»2ocH Ox. IEI = lzi- , Arg E = 2Arg z, s oco6oi1T04Kez=0 pacnoJJ01KeH ;:i:imonb.n11H11HM11 HBJJHIOTCH OKPY)f(HOCTH c ueHTpOM Ha oc11Ko~rnneKCHbIH rroTeHu11an s 3ana4ax :rneKrpocrnT11K11 ttaXOJll1TCH TaK)!(C, KaK 11 B 3a}la'!aX fl1JlPOJll1113.M11Kl1, T. e.
KaK cpyttKUl1H, OT06pa)f(at0maH3anaHHYIO o6JJaCTb Ha CTaHnapTHYIO o6JJaCTh, COOTBeTcrny1omy10 Tl1llYpaccMarp11saeMoi1 33.)13.411 (nonynnOCKOCTh, llOJJOCa MelKJlY napanneJJbHblMl1 npHMhlM11, BHeIIIHOCTh Kpyra, KOJlhUO).EcJJ11 a - nont0c.f{z) =fc_,,(z-a)TO 4neH 2 qi In -1-z- a(z) , 11.fCz) HMeeT B OKpecrnocT11 T04Kl1 a pa3JJO)!(eH11e,pi? . , n--+1c0 + c1(-.(. -a )+ .. .,+ ... +--+_qiz-az-aonpe}lenHeT s T04Ke a nnocK11if T04e4HbJH 3ap5!Jl se-n11411Hoi12q, o6a3Ha'!aeMhIH (a, 2q), 4neH pil(-::.-a)onpenenHeT s TO'!Kea .l.111ll0Jlh c MOMeHTOM p, o603Ha4aeMhlH (a, p).Ecntt Ha 6ecKoHe'IHOCTl1 KOMnneKcHhJH rroTeHu11anf(z) = C,,::"c+ ... + piz + 2qiln z + C0 +--1 + .. .,zTO 4JJeH 2qi In z orrpenenHeT Ha 6ecK0He'IHOCT11 nnocK11i1 T04e4HbIH 3ap»nBeJll1411HOH p = 2q, '!JleH3.piz onpe.l(eJ15IeT JlHllOJ!h c MOMeHTOM p.Hai1T11 3aKOH 113MeHeHHH T04e<rnoro 3ap»na (a, 2q) 11 Jll1ITOJJ5I (a, p)np11 0}lHOJJl1CTHOM KOHcpOpMHOM orn6pa)f(eHl111.352I1PI1JI0)!(EHI151 K MEXAHI1KE I1 <DI13l1KEOnwem.
Be1rnqttHa Toqeqttoro JapHp,a coxpattHeTcH, ecJIH KOHcpopMttoeOT06pa)!(eHHe B OKpecTHOCTH TQqKH a HMeeT BHA* 0,~(z) =a+ C 1(z - a)+ ... C 1TO(a, p) --7 (a, pC 1).4. I1oKa3aTb, qTQ KOMnJieKCHbltt noTeHuttaJI :rneKTPOCTaTHqecKoro CTOJIH,o6pa30B3HHOro ToqeqHbIM 33pHAOM (a, 2q) B npOH3BOJlbHOtt OAHOCBH3HOHo6nacTHD, onpep,enHeTcH cpopMynoil:f(z)=2qiln-1-+c,w(z, a)rp,e Ctl(Z, a) - cpyHKUHH, KOHcpOpMHO OT06pmKal()IU3H o6rraCTb D Ha e.[(HHHqHbltt Kpyr TaK, qro Ctl(a, a) =0,c - .ueil:CTBHTeJibH35! CTOCT0.5!HHa.5!.5. HattnI KOMnrreKCHhie noTeHUttaJihI :rneKTpocTaTttqecKttX norrett, o6pa3oBaHHhJX Jap,aHHhJMH roqeqHhJMH 1ap»AaM11 B YKaJaHHhIX 06rracr51x:a)Im z > 0, JapH.U (z0 , 2q);x3ap5!.U (z0 , 2q);l2r) ----,-+2>1, JapHA (=, 2q);B) lzl > R, 3apHA (zo, 2q);A) BHernHOCTh OTPeJKa< R,6) lzlba-µ-1<R,y=0,3apHA ( 00 , 2q).Onwembl.z-z0a) .f(z) = 2qi In - + C;6) .f(z) = 2qi Inz-zos) .f(z) = 2qi InR2-z z0R2-z z0z0 )R(z -+ C;+ C;R(z-z 0 ).1r).f(z)=2qzln--+C,w(z)1n).f(z) = 2qi In - - + C,w(z)w(z)=m(z) =z-~a-b,,,,c-=a--b-;z-~ .R6.
,UoKa3aTb, qTo 3JieKTpocrnn1qec1<oe none, o6pa3osatttt{)e n11norreM(a, p) B npOH3BOJlbHOtt OAHOCB513HOH o6JiaCTH D, onpenen5!eTCH KOMnJieKCHbJM n0Tettu11arr0Mj(z), rne cpyttKUHHj(z) orn6pa)!(aeT o6rracTh D Ha stternHOCTh rop1130HTaJibHoro OTPeJKa TaK, qTo f(a) =Toq1<e z=a pastta pil(z - a),ecn11a*00 ,tt pasttaoo11 rnaBHa5! qacTb .f(z) Bpiz,ecntta==.7. ITocTPOHTh 3rreKTPOCTaT11qecK11e noJI5!, 06pa3osatttth1e 3anattHhIMHAHCTOJ15!MH:< R, AHIIOJib (a, p);6) lzl > R, AHIIOJib (a, p);µ-1<R,y=0, .UHIIOJib (=, p);a) lzlB) BHernHOCTb OTPeJKar)x27l+b2>1, AHIIOJib (oo, p).353ITP11JIO)f{EHllil K MEXAH11KE l1 <Dl13l1KEOnwembz.pip •i.R2a)j(z)=--+--.
+C,a'1-0,a =-=-,pz-a z-aaj(z)= pi - P! z + C, a= 0, C zR•..6).f(z)=•R2 p=~,a.QeiicTBHTeJibHoe 'iHCJio;.-·___E!_+___E_!__+C ai'= f(7)=E!:__l!.!:_z+Ca==·z - a z - a*''~z R2'''s)j{z)= p(zcosa+ isin a.J z 2 -r)j(z)= _l!_[(az-b~z 2 -C 2 )cosa-i(bz -a~z 2 -C 2 )sin a]+ cl,R2 ) + C, pi= pe;a;a-bc2 =a2-b2.IInocKaJl 3a.n:aqa o crntt.n:apTHOM pacnpe.QeJieHHH TeMnepaTypbI BHyrpHTeJia xapaKTepH3yeTCJl aHam1rnqecKoi1 ¢YHKUHei1 j{z) =u(x, y) +iv(x, y),Ha3bIBaeMOH KOAtnJICKCHblA'l nome1-114uaJlOAt menJL06020 110Jlfl, ¢YHKUHJIu(x, y) - ¢YHKUHJI TeMnepaTyphI B TO'!Ke (x, y).
BeKTOpa= -k grad u= =-kf'(z)na3bIBaeTcR 6eKmopoAt nomoKa menlla,k -K03cpcp111.(11CHT Terrnorrpo-BO.D:HOCTI:I.IIoTOK Terrna qepe3 KOHTYP C paseHOKpeCTHOCTH TO'iKHa= -k fcdv=ik fcf'(z)dz.Ecmr Ba~Iqlj{z)= ... +--+C0 +C1(z-a)+--In--,z-aTO 'iJieH8.q1--In-2trk z-a2trkz-aorrpe.n:eJIJieT B TO'!Ke a HCTO'iHHK (a, q) o6HJibHOCTH q.HattTH pacrrpe.n:eJieHHe TeMrrepaTyphl s YKaJaHHbIX o6nacrnx no 3a-.n:aHHblM HCTO'iHHKaM, C'iHTaJl, 'ITO Ha rpam1ue TeMrrepaTypa IlOCTOJIHHa:a)Imz> 0,HCTO'iHHK (a, q);B) IlOJIYITOJIOCa ~I6) lzl< R,<a, y > 0,HCTO'iHHK= _q_Inlz-al+C;6) u(x, y)HCTO'iHHK (a, q);(ih, q), h > 0.Onrnembt.2a) u(x, y)2trz-a. trz .
h trhsm-+is-s) u(x, y)=_q_ln2tr2a2~ +C.. trz . h -trsm--is2a2a= _q_Inl R2tr-az l+C;R(z-a)CIIIICOK JIIITEP A TYPh l1. Ax11e1ep H. 11. JieKu1111 no Teop1111 3nnpoKCl1M3Ul111. M.: H3yI<a, 1965.2. 611u3me AB. OcHOBhI Teop1111 3Ha.Jll1TWiecKMx QJ)'HKUl1H . M .: H3)'K3, 1984.3. D11U3.[\3e A. B ., K3Jil1Hl1'1eHKO<D. C6opH11K 33.[\3'1 no yp3BHeHl1HMM3TeM3TH'leCKOH cp11111K11. M.: H3)'K3, 1977 .4. BOJIKOBb!CKl1H JI.
11., JiyHU r. JI., Ap3M3HOBl1'1 I1. r. C6opH11K 33.[\3'1no Teop1111 cpyHKU11H KOMnJieKCHoro nepeMeHHOro . M.: H3)'K3, 1975.n.5. .LJ:JKeHKl1HC .LJ:JK.0.D,HOJil1CTHbie cp)'HKUl111 11 KOHcpOpMHbie OT06p31KeHi'l.H.M.: MJI, 1962.6. Esrp3cpos M.A. AH3JIHTH4eCK11e cpyHKU11H. M.: H3)'K3, 1968.7. EBrp3cpOB M. A. Ac11M11TOTH'lecK11e oueHK11 11 ueJib1e cpyHKU11H. M.:H3)'K3, 1979.8. K3pTaH A 3JieMeHrnpH35! Teop11H 3HaJI11T11'lecK11x cpyttKu11i1 o.uttoro 11HeCKOJihKHX KOM11JieKCHhIX 11epeMeHHhIX. M.: MJI, 1963.9.
KyCHc IT. BBenettne B Teopmo npoCTp3HCTB Hp. M.: M11p, 1984.10.JI3BpettneB M. A., W363T 6. B . MeTOL\hI Teop1111 cp)'HKU11H KOM-nJieKcttoro 11epcMettttoro. M.: H3)'K3, 1987.11. JI3BpeHTbeB M. A., W363T I>. B. ITpo6JICMhI rnnpomrn3Ml1K11 11 11xMaTcMaT11YecK11e MO,llen11. M.: IIayKa, 1977.12.
JieoHTbeB A. <D. Uenb1e cjJyHKU1111, pHL\hI 3KCl10HeHT. M .: H3)'K3, 1983.13. JieoHThCB3 T. A. JieKu1111 no Teopm-1 cpyHKU11H KOM11neKcttoro nepeMCHHOro. M.: H3)"1HbJH MHp, 2004.14. J1eoHThCB3 T. A., l13HcpepoB B. C., CepoB B. C. 33Ll3'111 110 Teop1111cjiyHKU11H KOMrrneKCHoro rrepeMeHttoro. l11.u-Bo MfY, 1992.15. M3pKywes11q A. I1.
Teop1151 3H3Jil1Tl14CCKl1X cpyHKUHH. T. 1, 2. M.:H3)'K3, 1967.16. H11KOJlhCK11i1 C. M. Ilp116n111KeH11e QJ)'HKUHH MHornx rrepeMeHHhIX 11TeopeMbI BJlOJKCHl15l. M.: H3)'K3, 1969.17. l10JII13 r., Cere r. 33.U3'11111 TeopeMbJ 3HaJIH33. T. 1, 2. M.: H3)'K3, 1978.18. l1pHB3JlOB I1.
I1. Bse.ueH11e s Teopmo cpyHKUHH KOM11JieKcHoro 11epeMeHHoro. M.: H3)'K3, 1984.19. C6opmrn 33L\3'1 110 Teop1111 3H3JI11rnqecK11x cpyHKUHH I 11on pen.M. A. Esrp3cpos3. M .: H3)'K3, 1974.20. C11nopoB IO. B., <De.uop10K M. B., lll36ytt11H M. I1. J1eKu1111 110 Teop1111cpyttKUHH KOMnJieKcHoro rrepeMeHHoro.
M.: H3)'K3, 1982.21. Crn11noB C. Teop1151 cpyHKUMH KOM11neKcttoro rrepeMeHHoro. T. 1, 2. M.:MJI, 1962.22. T11xoM11pos B. M. HeKoTOpb1e sorrpocb1 Teop1111 rrp116JI111KeH11i1. M.:lfa.[(-BO MfY, 1976.23. <De.uop!OK M. B. Ac11MIITOTHK3: HHTerp3JlhI 11 p5!.UhL M.: H3)'K3, 1987.24. ill363T I>. B. Bse;i:eH11e B KOMIIJieKCHhIH 3HaJII13. T. 1, 2. M.: HayI<a, 1976.IlPE,I(ME TH hillA6cm1cca cxo1111MocT11, 126ABTOCOnpmKeHHOe ypaBHeH11enpllMOH, 11A-TO'-!Ka, 101nopllllOK, KpaTHOCThA-T04Kl1, 101Attan11T11'-!eCKOe npo11onlKem1ecjJyHKU1111, 138HenocpencTBeHHoe, 138B eKTop noToKa Ternia, 353B11xpe11cTO'-IHl1K 0611JibHOCT11 N 1111HTeHCJ1BHOCTJ1 f,345,347BhmyKJiall MalKopattrn, 249BbI'-leT, 157rapMOHl1'-leCKl1e CTOJIHHOMbl, 209.LJ:11nOJib, 347.IJ:t1cJ:icpcpe111.1t1pyeMOCTb cpyttKL\Htt, 553ana'-la.LJ:11p11xne, 196Kourn, 219HewMaHa, 196llttn11KaTp11ca poCTa, 125IifHTcrpanI>eccenll, 118.LJ:11p11xne, 80, 208Jlannaca, 77paBHOMepHO CXO.Z:Vlllll1HCll, 71Tttna Kourn, 83<l>peHeJill, 80IifHTerpanbHaa cjJopMyna Kourn, 83I1HTerp11posatt11e cjlyttKUl1H, 67IifHTepnonau110HHblH pan HblOTOHa,125Ii!CTO'-IHHK, 346YKA3ATE.JlhKoMnneKcHoe '-!11cno, 5Monynh, 5Tp11rottoMeTp11'-lecKaa cjJopMa, 6KoMnneKcHo-conpalKemrne '-IHCJIO.
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